Contextual integral models | Year 12

01 / LEARN

The idea, step by step

Define the quantity before integrating

Name the dependent quantity and explain what its derivative measures. Use the physical start and end values as bounds, rather than automatically using time 0. For a marginal-cost function, the independent variable is output quantity; for a flow function it is time.

A model answer should contain the setup, calculation, units and a contextual sentence. If a rate is negative, decide whether this means loss, reverse movement, or a model that is outside its valid range. State any needed assumptions such as continuous output or unchanged operating conditions.

Average value is total divided by interval length

The average value of a continuous function over $[a, b]$ is $\frac{1}{b - a} \int_{a}^{b} f (x) d x$ . It is the height of a rectangle with the same width and signed area. It is not generally the mean of the two endpoint values.

In a report, distinguish an exact answer within the mathematical model from a physically exact prediction. Measurement error, rounded parameters and simplifying assumptions can all limit the precision of the final claim.

f_{average} = \frac{1}{b - a} \int_{a}^{b} f (x) d x

A common trap

Calling an average rate the total, or averaging only the endpoint values for a curved function.

Instead: Identify whether the question asks for total, change, final amount or average; use the corresponding integral relationship.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

A product has marginal cost $C^{'} (q) = 0.03 q^{2} + 2$ dollars/item and fixed cost $C (0) = 100$ dollars. Find $C (10)$ .

Additional cost $= \int_{0}^{10} (0.03 q^{2} + 2) d q$ .
${[0.01 q^{3} + 2 q]}_{0}^{10} = 10 + 20 = 30$ .
Add fixed cost 100; the model treats output as continuous.

130 dollars

Example 2

Worked example

Find the average value of $f (t) = t^{2}$ on $[0, 3]$ .

Total $= \int_{0}^{3} t^{2} d t = 9$ .
Divide by interval length 3, giving 3.
The endpoint average is 4.5, which is a different quantity.

3

Check your understanding

A rate is $2 + t$ L/min on $[0, 4]$ . Find the average rate.

Show answer

$4$ L/min

Total $= {[2 t + t^{2} / 2]}_{0}^{4} = 16$ L; divide by 4 min.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the average value of $f (x) = 3 x^{2}$ on $[0, 2]$ .
3 marks
$C^{'} (q) = 2 q + 5$ dollars/item and $C (0) = 40$ . Find $C (6)$ .
3 marks
Water flows at $12 e^{- t}$ L/min for $0 \leq t \leq ln 2$ . Find the supplied volume.
3 marks
A force $F (x) = 4 x$ newtons moves an object from $x = 1$ to $x = 3$ metres. Find $\int_{1}^{3} F (x) d x$ and its unit.
3 marks
A rate model $r (t) = 10 - 2 t$ items/hour is used for production until $t = 8$ hours. Explain a limitation.
3 marks
The average value of $f (x) = k x$ on $[0, 4]$ is 6. Find $k$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the average value of $f (x) = 3 x^{2}$ on $[0, 2]$ .
3 marks
The integral is 8; divide by interval length 2.
$4$
$C^{'} (q) = 2 q + 5$ dollars/item and $C (0) = 40$ . Find $C (6)$ .
3 marks
Change $= {[q^{2} + 5 q]}_{0}^{6} = 36 + 30 = 66$ .
Add 40.
106 dollars
Water flows at $12 e^{- t}$ L/min for $0 \leq t \leq ln 2$ . Find the supplied volume.
3 marks
${[- 12 e^{- t}]}_{0}^{ln 2} = - 6 + 12 = 6$ .
$6$ L
A force $F (x) = 4 x$ newtons moves an object from $x = 1$ to $x = 3$ metres. Find $\int_{1}^{3} F (x) d x$ and its unit.
3 marks
${[2 x^{2}]}_{1}^{3} = 18 - 2 = 16$ N·m.
One newton-metre is one joule.
16 joules
A rate model $r (t) = 10 - 2 t$ items/hour is used for production until $t = 8$ hours. Explain a limitation.
3 marks
The zero is $t = 5$ ; a revised model or valid interval is needed beyond that time.
The model predicts negative production after 5 hours.
The average value of $f (x) = k x$ on $[0, 4]$ is 6. Find $k$ .
3 marks
$\frac{1}{4} \int_{0}^{4} k x d x = 2 k$ .
Solve $2 k = 6$ .
$k = 3$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods integration and logarithmic antiderivatives; original learning and assessment material, with by-hand methods and explicitly described numerical models.

Official curriculum source ↗

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